Periods of singular double octic Calabi-Yau threefolds and modular forms

TL;DR

This paper explores the relationship between period integrals of certain Calabi-Yau threefolds and modular forms, providing numerical evidence for a Mellin transform-based interpretation in the case of double octics.

Contribution

It proposes a new interpretation of period integrals of nodal Calabi-Yau threefolds using Mellin transforms of associated modular forms, supported by numerical evidence.

Findings

Numerical evidence supports the Mellin transform interpretation.

Period integrals relate to special values of modular forms.

Application to double octic Calabi-Yau threefolds.

Abstract

By the modularity theorem every rigid Calabi-Yau threefold $X$ has associated modular form $f$ such that the equality of $L$-functions $L(X,s)=L(f,s)$ holds. In this case period integrals of $X$ are expected to be expressible in terms of the special values $L(f,1)$ and $L(f,2)$. We propose a similar interpretation of period integrals of a nodal model of $X$. It is given in terms of certain variants of a Mellin transform of $f$. We provide numerical evidence towards this interpretation based on a case of double octics.